Lecture 14: Single Interface Reflection and Transmission

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Lecture 14

Single Interface Reflection and

Transmission

Much of the contents of this lecture can be found in Kong, and also the ECE 350X lecture

notes. They can be found in many textbooks, even though the notations can be slightly

different [29, 31, 38, 47, 48, 59, 71, 75, 77, 78].

14.1 Reflection and Transmission—Single Interface Case

We will derive the reflection coefficients for the single interface case. These reflection co-

efficients are also called the Fresnel reflection coefficients because they were first derived by

Austin-Jean Fresnel (1788-1827). Note that he lived before the completion of Maxwell’s equa-

tions in 1865. But when Fresnel derived the reflection coefficients in 1823, they were based

on the elastic theory of light; and hence, the formulas are not exactly the same as what we

are going to derive (see Born and Wolf, Principles of Optics, p. 40 [86]).

The single-interface reflection and transmission problem is homomorphic to the transmis-

sion line problem, albeit with complicated mathematics, as we have to keep track of the 3D

polarizations of the electromagnetic fields in this case. We shall learn later that the mathe-

matical homomorphism can be used to exploit the simplicity of transmission line theory in

seeking the solutions to the multiple interface problems.

133

134 Electromagnetic Field Theory

14.1.1 TE Polarization (Perpendicular or E Polarization)1

Figure 14.1: A schematic showing the reflection of the TE polarization wave impinging on a

dielectric interface.

To set up the above problem, the wave in Region 1 can be written as Ei + Er . We assume

plane wave polarized in the y direction where the wave vectors are βi = ˆxβix + ˆzβiz , βr =

ˆxβrx − ˆzβrz , βt = ˆxβtx + ˆzβtz , respectively for the incident, reflected, and transmitted waves.

Then

Ei = ˆyE0e−jβi·r = ˆyE0e−jβixx−jβiz z (14.1.1)

and

Er = ˆyRT E E0e−jβr ·r = ˆyRT E E0e−jβrxx+jβrz z (14.1.2)

In Region 2, we only have transmitted wave; hence

Et = ˆyT T E E0e−jβt·r = ˆyT T E E0e−jβtxx−jβtz z (14.1.3)

In the above, the incident wave is known and hence, E0 is known. From (14.1.2) and (14.1.3),

RT E and T T E are unknowns yet to be sought. To find them, we need two boundary conditions

to yield two equations.2 These are tangential E continuous and tangential H continuous,

which are ˆn × E continuous and ˆn × H continuous conditions at the interface.

1These polarizations are also variously know as the s and p polarizations, a descendent from the notations

for acoustic waves where s and p stand for shear and pressure waves respectively.

2Here, we will treat this problem as a boundary value problem where the unknowns are sought from

equations obtained from boundary conditions.

Single Interface Reflection and Transmission 135

Imposing ˆn × E continuous at z = 0, we get

E0e−jβixx + RT E E0e−jβrxx = T T E E0e−jβtxx, ∀x (14.1.4)

In order for the above to be valid for all x, it is necessary that βix = βrx = βtx, which is also

known as the phase matching condition.3 From the above, by letting βix = βrx = β1 sin θi =

β1 sin θr , we obtain that θr = θi or that the law of reflection that the angle of reflection is

equal to the angle of incidence. By letting βtx = β2 sin θt = βix = β1 sin θi, we obtain Snell’s

law that β1 sin θi = β2 sin θt. (This law of refraction that was also known in the Islamic world

in the 900 AD. [87]). Now, canceling common terms on both sides of the equation (14.1.4),

the above simplifies to

1 + RT E = T T E (14.1.5)

To impose ˆn × H continuous, one needs to find the H field using ∇ × E = −jωμH, or

that H = −jβ × E/(−jωμ) = β × E/(ωμ). By so doing

Hi = βi × Ei

ωμ1

= βi × ˆy

ωμ1

E0e−jβi·r = ˆzβix − ˆxβiz

ωμ1

E0e−jβi·r (14.1.6)

Hr = βr × Er

ωμ1

= βr × ˆy

ωμ1

RT E E0e−jβr ·r = ˆzβrx + ˆxβrz

ωμ2

RT E E0e−jβr ·r (14.1.7)

Ht = βt × Et

ωμ2

= βt × ˆy

ωμ2

T T E E0e−jβt·r = ˆzβtx − ˆxβtz

ωμ2

T T E E0e−jβt·r (14.1.8)

Imposing ˆn × H continuous or Hx continuous at z = 0, we have

βiz

ωμ1

E0e−jβixx − βrz

ωμ1

RT E E0e−jβrxx = βtz

ωμ2

T T E E0e−jβtxx (14.1.9)

As mentioned before, the phase-matching condition requires that βix = βrx = βtx. The

dispersion relation for plane waves requires that

β2

ix + β2

iz = β2

rx + β2

rz = ω2μ1ε1 = β2

1 (14.1.10)

β2

tx + β2

tz = ω2μ2ε2 = β2

2 (14.1.11)

Since βix = βrx = βtx = βx, the above implies that βiz = βrz = β1z . Moreover, βtz = β2z 6 =

β1z usually since β1 6 = β2. Then (14.1.9) simplifies to

β1z

μ1

(1 − RT E ) = β2z

μ2

T T E (14.1.12)

where β1z = √β2

1 − β2

x, and β2z = √β2

2 − β2

3The phase-matching condition can also be proved by taking the Fourier transform of the equation with

respect to x. Among the physics community, this is also known as momentum matching, as the wavenumber

of a wave is related to the momentum of the particle.

136 Electromagnetic Field Theory

Solving (14.1.5) and (14.1.12) yields

RT E =

( β1z

μ1

− β2z

μ2

) / ( β1z

μ1

+ β2z

μ2

)

(14.1.13)

T T E = 2

( β1z

μ1

) / ( β1z

μ1

+ β2z

μ2

)

(14.1.14)

14.1.2 TM Polarization (Parallel or H Polarization)

Figure 14.2: A similar schematic showing the reflection of the TM polarization wave impinging

on a dielectric interface. The solution to this problem can be easily obtained by invoking

duality principle.

The solution to the TM polarization case can be obtained by invoking duality principle where

we do the substitution E → H, H → −E, and μ ε as shown in Figure 14.2. The reflection

coefficient for the TM magnetic field is then

RT M =

( β1z

ε1

− β2z

ε2

) / ( β1z

ε1

+ β2z

ε2

)

(14.1.15)

T T M = 2

( β1z

ε1

) / ( β1z

ε1

+ β2z

ε2

)

(14.1.16)

Please remember that RT M and T T M are reflection and transmission coefficients for the

magnetic fields, whereas RT E and T T E are those for the electric fields. Some textbooks may

define these reflection coefficients based on electric field only, and they will look different, and

duality principle cannot be applied.

Single Interface Reflection and Transmission 137

14.2 Interesting Physical Phenomena

Three interesting physical phenomena emerge from the solutions of the single-interface prob-

lem. They are total internal reflection, Brewster angle effect, and surface plasmonic resonance.

We will look at them next.

14.2.1 Total Internal Reflection

Total internal reflection comes about because of phase matching (also called momentum

matching). This phase-matching condition can be illustrated using β-surfaces (same as k-

surfaces in some literature), as shown in Figure 14.3. It turns out that because of phase

matching, for certain interfaces, β2z becomes pure imaginary.

Figure 14.3: Courtesy of J.A. Kong, Electromagnetic Wave Theory [31]. Here, k is synony-

mous with β. Also, the x axis is equivalent to the z axis in the previous figure.

138 Electromagnetic Field Theory

Figure 14.4: Courtesy of J.A. Kong, Electromagnetic Wave Theory. Here, k is synonymous

with β, and x axis is the same as our z axis.

As shown in Figures 14.3 and 14.4, because of the dispersion relation that β2

rx + β2

rz =

β2

ix + β2

iz = β2

1 , β2

tx + β2

tz = β2

2 , they are equations of two circles in 2D whose radii are β1

and β2, respectively. (The tips of the β vectors for Regions 1 and 2 have to be on a spherical

surface in the βx, βy , and βz space in the general 3D case, but in this figure, we only show a

cross section of the sphere assuming that βy = 0.)

Phase matching implies that the x-component of the β vectors are equal to each other

as shown. One sees that θi = θr in Figure 14.4, and also as θi increases, θt increases. For

an optically less dense medium where β2 < β1, according to the Snell’s law of refraction, the

transmitted β will refract away from the normal, as seen in the figure. Therefore, eventually

the vector βt becomes parallel to the x axis when βix = βrx = β2 = ω√μ2ε2 and θt = π/2.

The incident angle at which this happens is termed the critical angle θc.

Since βix = β1 sin θi = βrx = β1 sin θr = β2, or

sin θr = sin θi = sin θc = β2

β1

√μ2ε2

√μ1ε1

= n2

(14.2.1)

where n1 is the reflective index defined as c0/vi = √μiεi/√μ0ε0 where vi is the phase velocity

of the wave in Region i. Hence,

θc = sin−1(n2/n1) (14.2.2)

When θi > θc. βx > β2 and β2z = √β22 − βx2 becomes pure imaginary. When β2z

becomes pure imaginary, the wave cannot propagate in Region 2, or β2z = −jα2z , and the

Single Interface Reflection and Transmission 139

wave becomes evanescent. The reflection coefficient (14.1.13) becomes of the form

RT E = (A − jB)/(A + jB) (14.2.3)

It is clear that |RT E | = 1 and that RT E = ejθT E . Therefore, a total internally reflected wave

suffers a phase shift. A phase shift in the frequency domain corresponds to a time delay in

the time domain. Such a time delay is achieved by the wave traveling laterally in Region 2

before being refracted back to Region 1. Such a lateral shift is called the Goos-Hanschen shift

as shown in Figure 14.5 [86]. A wave that travels laterally along the surface of two media is

also known as lateral waves [88, 89].

Please be reminded that total internal reflection comes about entirely due to the phase-

matching condition when Region 2 is a faster medium than Region 1. Hence, it will occur

with all manner of waves, such as elastic waves, sound waves, seismic waves, quantum waves

etc.

Figure 14.5: Goos-Hanschen Shift. A phase delay is equivalent to a time delay (courtesy of

Paul R. Berman (2012), Scholarpedia, 7(3):11584 [90]).

The guidance of a wave in a dielectric slab is due to total internal reflection at the dielectric-

to-air interface. The wave bounces between the two interfaces of the slab, and creates evanes-

cent waves outside, as shown in Figure 14.6. The guidance of waves in an optical fiber works

by similar mechanism of total internal reflection, as shown in Figure 14.7. Due to the tremen-

dous impact the optical fiber has on modern-day communications, Charles Kao, the father of

the optical fiber, was awarded the Nobel Prize in 2009. His work was first published in [91].

140 Electromagnetic Field Theory

Figure 14.6: Courtesy of E.N. Glytsis, NTUA, Greece [92].

Figure 14.7: Courtesy of Wikepedia [93].

Waveguides have affected international communications for over a hundred year now.

Since telegraphy was in place before the full advent of Maxwell’s equations, submarine cables

for global communications were laid as early as 1850’s. Figure 14.8 shows a submarine cable

from 1869 using coaxial cable,, and one used in the modern world using optical fiber.

Single Interface Reflection and Transmission 141

Figure 14.8: The picture of an old 1869 submarine cable made of coaxial cables (left), and

modern submarine cable made of optical fibers (right) (courtesy of Atlantic-Cable [94], and

Wikipedia [95].

Bibliography

[1] J. A. Kong, Theory of electromagnetic waves. New York, Wiley-Interscience, 1975.

[2] A. Einstein et al., “On the electrodynamics of moving bodies,” Annalen der Physik,

vol. 17, no. 891, p. 50, 1905.

[3] P. A. M. Dirac, “The quantum theory of the emission and absorption of radiation,” Pro-

ceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical

and Physical Character, vol. 114, no. 767, pp. 243–265, 1927.

[4] R. J. Glauber, “Coherent and incoherent states of the radiation field,” Physical Review,

vol. 131, no. 6, p. 2766, 1963.

[5] C.-N. Yang and R. L. Mills, “Conservation of isotopic spin and isotopic gauge invari-

ance,” Physical review, vol. 96, no. 1, p. 191, 1954.

[6] G. t’Hooft, 50 years of Yang-Mills theory. World Scientific, 2005.

[7] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation. Princeton University

Press, 2017.

[8] F. Teixeira and W. C. Chew, “Differential forms, metrics, and the reflectionless absorp-

tion of electromagnetic waves,” Journal of Electromagnetic Waves and Applications,

vol. 13, no. 5, pp. 665–686, 1999.

[9] W. C. Chew, E. Michielssen, J.-M. Jin, and J. Song, Fast and efficient algorithms in

computational electromagnetics. Artech House, Inc., 2001.

[10] A. Volta, “On the electricity excited by the mere contact of conducting substances

of different kinds. in a letter from Mr. Alexander Volta, FRS Professor of Natural

Philosophy in the University of Pavia, to the Rt. Hon. Sir Joseph Banks, Bart. KBPR

S,” Philosophical transactions of the Royal Society of London, no. 90, pp. 403–431, 1800.

[11] A.-M. Amp`ere, Expos´e m´ethodique des ph´enom`enes ´electro-dynamiques, et des lois de

ces ph´enom`enes. Bachelier, 1823.

[12] ——, M´emoire sur la th´eorie math´ematique des ph´enom`enes ´electro-dynamiques unique-

ment d´eduite de l’exp´erience: dans lequel se trouvent r´eunis les M´emoires que M.

Amp`ere a communiqu´es `a l’Acad´emie royale des Sciences, dans les s´eances des 4 et

189

190 Electromagnetic Field Theory

26 d´ecembre 1820, 10 juin 1822, 22 d´ecembre 1823, 12 septembre et 21 novembre 1825.

Bachelier, 1825.

[13] B. Jones and M. Faraday, The life and letters of Faraday. Cambridge University Press,

2010, vol. 2.

[14] G. Kirchhoff, “Ueber die aufl¨osung der gleichungen, auf welche man bei der unter-

suchung der linearen vertheilung galvanischer str¨ome gef¨uhrt wird,” Annalen der Physik,

vol. 148, no. 12, pp. 497–508, 1847.

[15] L. Weinberg, “Kirchhoff’s’ third and fourth laws’,” IRE Transactions on Circuit Theory,

vol. 5, no. 1, pp. 8–30, 1958.

[16] T. Standage, The Victorian Internet: The remarkable story of the telegraph and the

nineteenth century’s online pioneers. Phoenix, 1998.

[17] J. C. Maxwell, “A dynamical theory of the electromagnetic field,” Philosophical trans-

actions of the Royal Society of London, no. 155, pp. 459–512, 1865.

[18] H. Hertz, “On the finite velocity of propagation of electromagnetic actions,” Electric

Waves, vol. 110, 1888.

[19] M. Romer and I. B. Cohen, “Roemer and the first determination of the velocity of light

(1676),” Isis, vol. 31, no. 2, pp. 327–379, 1940.

[20] A. Arons and M. Peppard, “Einstein’s proposal of the photon concept–a translation of

the Annalen der Physik paper of 1905,” American Journal of Physics, vol. 33, no. 5,

pp. 367–374, 1965.

[21] A. Pais, “Einstein and the quantum theory,” Reviews of Modern Physics, vol. 51, no. 4,

p. 863, 1979.

[22] M. Planck, “On the law of distribution of energy in the normal spectrum,” Annalen der

physik, vol. 4, no. 553, p. 1, 1901.

[23] Z. Peng, S. De Graaf, J. Tsai, and O. Astafiev, “Tuneable on-demand single-photon

source in the microwave range,” Nature communications, vol. 7, p. 12588, 2016.

[24] B. D. Gates, Q. Xu, M. Stewart, D. Ryan, C. G. Willson, and G. M. Whitesides, “New

approaches to nanofabrication: molding, printing, and other techniques,” Chemical

reviews, vol. 105, no. 4, pp. 1171–1196, 2005.

[25] J. S. Bell, “The debate on the significance of his contributions to the foundations of

quantum mechanics, Bells Theorem and the Foundations of Modern Physics (A. van

der Merwe, F. Selleri, and G. Tarozzi, eds.),” 1992.

[26] D. J. Griffiths and D. F. Schroeter, Introduction to quantum mechanics. Cambridge

University Press, 2018.

[27] C. Pickover, Archimedes to Hawking: Laws of science and the great minds behind them.

Oxford University Press, 2008.

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